Because of the numerous applications where structured rational and polynomial matrices occur, there have been many attempts to construct
linearizations for such structured rational and polynomial matrices that display the same structure as that of the rational or polynomial matrix (see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]} among many other references on this topic). An important motivation for this search is to preserve numerically in floating point arithmetic the symmetries of the zeros and poles of these structured problems by applying structured algorithms for structured generalized eigenvalue problems to these structured linearizations [10]}, [11]}, [12]}, [13]}, [14]}, [15]}. However, all these earlier attempts to construct structured linearizations find obstacles when they are applied to the structured problems considered in this paper, because they either cover only a subclass of the structures, or they impose certain conditions on the rational and polynomial matrices for their construction to apply, such as regularity, strict properness or invertibility of certain matrix coefficients. We emphasize that, for some polynomial matrices, the mentioned obstacles cannot be overcome in any way with the previously adopted definitions of linearization, because it has been proved in [9]}, [17]} that there exist alternating polynomial matrices which cannot be linearized at all according to the standard definitions of linearizations in [9]}, [17]}. In contrast, in the present paper, we give a construction of structured strongly minimal linearizations valid for arbitrary rational and polynomial matrices, with any of the above four structures. Moreover, the proof used for this construction is different from these earlier papers, and we claim it to be simpler as well.
